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Worksheet: Using Quadratic Equations to Solve Word Problems

In this worksheet, we will practice solving word problems by forming and solving quadratic equations.

Q1:

The length of a rectangle is 2 cm more than its width. If its area is 80 cm2, what are its length and width?

A40 cm, 8 cm

B40 cm, 2 cm

C40 cm, 10 cm

D10 cm, 8 cm

Q2:

The length of a rectangle is 4 cm more than its width. If its area is 96 cm2, what are its length and width?

A24 cm, 8 cm

B24 cm, 4 cm

C24 cm, 12 cm

D12 cm, 8 cm

Q3:

The length of a rectangle is 10 cm more than its width. If its area is 75 cm2, what are its length and width?

A7 cm, 5 cm

B7 cm, 10 cm

C7 cm, 15 cm

D15 cm, 5 cm

Q4:

The length of a rectangle is 2 cm more than its width. If its area is 48 cm2, what are its length and width?

A24 cm, 6 cm

B24 cm, 2 cm

C24 cm, 8 cm

D8 cm, 6 cm

Q5:

The length of a rectangle is 26 cm
more than its width. Given that its area is 120 cm2,
determine its perimeter.

Q6:

The length of a rectangle is 15 cm
more than its width. Given that its area is 54 cm2,
determine its perimeter.

Q7:

The length of a rectangle is 10 cm
more than its width. Given that its area is 24 cm2,
determine its perimeter.

Q8:

The side length of a square is 𝑥 cm,
and the dimensions of a rectangle are 𝑥 cm
and 2 cm. Given that the sum of their areas is 8 cm2, determine the perimeter of the square.

Q9:

The side length of a square is 𝑥 cm,
and the dimensions of a rectangle are 𝑥 cm
and 5 cm. Given that the sum of their areas is 14 cm2, determine the perimeter of the square.

Q10:

The side length of a square is 𝑥 cm,
and the dimensions of a rectangle are 𝑥 cm
and 8 cm. Given that the sum of their areas is 9 cm2, determine the perimeter of the square.

Q11:

The length of a rectangle is 3 cm more than double the width.
The area of the rectangle is 27 cm2.
Write an equation that can be used to find 𝑤, the width of the rectangle, in centimeters.

A𝑤(3𝑤+2)=27

B𝑤(𝑤+3)=27

C2𝑤(2𝑤+3)=27

D𝑤(2𝑤+3)=27

E𝑤(𝑤+2)=27

Q12:

A rectangular photograph measuring 6 cm
by 4 cm is to be displayed in a card mount in a rectangular frame,
as shown in the diagram.

Write an equation that can be used to find 𝑥, the width of the mount, if its area is
64 cm2.

A(7+2𝑥)(5+2𝑥)−13=64

B(3+2𝑥)(6−5𝑥)−21=32

C(9+5𝑥)(3+7𝑥)−15=18

D(4+2𝑥)(6+2𝑥)−24=64

E(5+2𝑥)(7−2𝑥)−24=28

Q13:

The sum, 𝑆, of the first 𝑛 consecutive integers (1+2+3+4+⋯+𝑛) can be found using
Starting from 1, how many consecutive integers are required to make a sum of 21?

Q14:

The height of a ball 𝑡 seconds after it was kicked from the ground is modeled by the
function ℎ, where ℎ(𝑡)=15𝑡−5𝑡2.

For how long does the ball remain in the air?

For how long does the ball remain above a height of 10 m?

Q15:

Which of the following exceeds its multiplicative inverse by 1130?

A−6 or 5

B−65 or 56

C
6
or −5

D65 or −56

Q16:

A study was carried out to determine how many people in a small town were infected with the
hepatitis C virus. An approximation for the number of infected people, 𝑦,
can be found using 𝑦=−0.5𝑛−5.5𝑛+9312, where 𝑛 is the number of years
after 2006. In which year do we expect
there to be no infected people?

Q17:

Find the solution set of the equation 𝑥−26𝑥9=−1692 in ℝ.

A{2,8}

B−2,−89

C{−2,−8}

D2,89

E−2,89

Q18:

Given that nine times the square of 𝑥 is 25, what are the possible values of 𝑥?

A35

B35 or −35

C5 or −5

D53 or −53

Q19:

Determine the positive number whose square exceeds twice its value by 15.

Q20:

Find the positive number whose square is equal to two times the number.

Q21:

The difference between the square of Daniel’s age now and 5 times his age 2 years ago is 160. How old is Daniel now?

Q22:

At which values of 𝑥
does the graph of 𝑦=12𝑥−8𝑥2 cross the 𝑥-axis?

A0 and 2

B0 and −23

C0 and 83

D0 and 23

E0 and −83

Q23:

When twice the square of a number is added to
1, the result is 99. What is the number?

A98

B10 or −10

C10

D7 or −7

Q24:

Find two numbers with a sum of 10 and a product of 9.

A8, 2

B5, 5

C3, 7

D1, 9

E6, 5

Q25:

A study was carried out to investigate the number of people in a town infected by
norovirus. The number of people infected, 𝑦, occurring
𝑛 years after the start of the study, can be found using the equation
What was the
value of 𝑛 when there were 347 people
infected?